
\(K s \in 4Z\)  [ \(SO(32)\)  ]
\(K s_1, K s_2 \in 4Z\)  or \(K s_1, K s_2 \in 4Z+2\)  [ \(E_8 \times E_8\)  ] :
In these cases the aspects are almost parallel to that of [1]}.
The constraint (REF ) or (REF ) implies
\(\sum _{i \in S_1}\, m_{i,L} + LM \equiv 0 ~ (\mbox{mod}\, 2),\) 
where \(S_1\)  has been defined in (REF ), that is,
\(S_1 \equiv \left\lbrace i\in S~:~ d_i \in 2Z+1\right\rbrace .\nonumber \) 
We then find
\(\sum _{\beta ^{\prime }} \, Z_{(\alpha ,2\beta ^{\prime })}(\tau ) \ne 0,\) 
since we generically possess many states satisfying the condition (REF ).
This means that \(\widehat{\delta }\) -orbifolding cannot satisfy (REF ) by itself.
However, as was shown in [1]}In comparison with [1]}, the roles of left and right movers have been exchanged here., we can make it possible
by further introducing the \(Z_2\) -orbifold action \(\widehat{\gamma }\) , which commutes with \(\widehat{\delta }\) ;
\(\widehat{\gamma }:= \left\lbrace \begin{array}{ll}(-1)^{F_L} \otimes \gamma _R& ~~ (\# S_1 \in 2Z+1), \\\gamma _R& ~~ (\# S_1 \in 2Z) ,\end{array}\right.\hspace{28.45274pt}\gamma _R := \prod _{i\in S_1}\, (-1)^{\ell _{i, R}}.\) 
on the right-moving minimal characters \(\displaystyle \overline{\prod _{i\in S} \mbox{ch}^{(\sigma )}_{\ell _{i,R}, m_{i,R}-2n_R}(\tau )}\) ,
and \((-1)^{F_L}\)  denotes the sign-flip of the left-moving R-sector.
We shall also introduce the discrete torsion
[4]}, [5]}, [6]}
with respect to the \(\widehat{\gamma }\)  and \(\widehat{\delta }\) -actions;
\(\xi \left( a, \alpha \, ; b ,\beta \right) := (-1)^{\left(LM-1\right) \left(a\beta - b \alpha \right)},\hspace{28.45274pt} \left(a, b \in Z_2, ~ \alpha , \beta \in Z_{4K} \right),\) 
where \(a,b\)  label the spatial and temporal twistings by \(\widehat{\gamma }\) , while \(\alpha ,\beta \)  are those associated to \(\widehat{\delta }\)  as above.
Then, for any fixed \( \alpha \in 2Z+1\) , we readily obtain
\(\sum _{\beta ^{\prime }} \, \left. Z_{(\alpha ,2\beta ^{\prime })}(\tau )\right|_{\widehat{\gamma }-\mbox{\scriptsize orbifold}} & \equiv \frac{1}{2} \sum _{\beta ^{\prime }}\, \sum _{a,b\in Z_2}\, \xi \left( a, \alpha \, ; b ,2\beta ^{\prime } \right)\, Z_{(a, \alpha ; \, b, 2\beta ^{\prime })} (\tau ) =0.\) 

That is, the criterion (REF ) is satisfied when combining the orbifolding by \(\widehat{\gamma }\)  and \(\widehat{\delta }\) .
See [1]} for more detailed arguments.

\(K s \in 4Z+2 \)  [\(SO(32)\) ]
\(K s_1 \in 4Z\) , \(K s_2 \in 4Z+2\) , or \(K s_1 \in 4Z+2\) , \(K s_2 \in 4Z\)  [\(E_8\times E_8\) ] :
In these cases, the constraint (REF ) or (REF ) yields
\(\sum _{i \in S_1}\, m_i + LM +1 \equiv 0 ~ (\mbox{mod}\, 2),\) 
in place of (REF ).
Therefore, we can make the criterion (REF ) to be satisfied by taking again the \(\widehat{\delta }\)  and \(\widehat{\gamma }\)  orbifolds
but with the different discrete torsion
\(\xi ^{\prime } \left( a, \alpha \, ; b ,\beta \right) := (-1)^{LM \left(a\beta - b \alpha \right)},\hspace{28.45274pt} \left(a, b \in Z_2, ~ \alpha , \beta \in Z_{4K} \right).\) 

\(K s \in 2Z+1 \)  [\(SO(32)\) ]
\(K(s_1+s_2) \in 2Z+1\)  [\(E_8\times E_8\) ] :
In these cases, any states cannot satisfy
the condition (REF ),
and thus the criterion (REF ) is trivially achieved
by only making the \(\widehat{\delta }\) -orbifolding.

\(K s_1 , K s_2 \in 2Z+1\)  [\(E_8\times E_8\) ] :
In these remaining cases, both of (REF ) and (REF ) are possible, depending on which theta function
factors \(({\theta }_j)^{s_1} ({\theta }_k)^{s_2}\)  the operator \(\widehat{\delta }\)  acts.
Thus, (REF ) cannot be satisfied even if incorporating the \(\widehat{\gamma }\) -orbifolding.
We conclude that the string vacua with the desired properties are not constructed in these cases.

